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Calculus Examples
Step 1
Step 1.1
Subtract from both sides of the equation.
Step 2
Step 2.1
Differentiate with respect to .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 3
Step 3.1
Differentiate with respect to .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Simplify the expression.
Step 3.6.1
Add and .
Step 3.6.2
Multiply by .
Step 4
Step 4.1
Substitute for and for .
Step 4.2
Since the left side does not equal the right side, the equation is not an identity.
is not an identity.
is not an identity.
Step 5
Step 5.1
Substitute for .
Step 5.2
Substitute for .
Step 5.3
Substitute for .
Step 5.3.1
Substitute for .
Step 5.3.2
Subtract from .
Step 5.3.3
Substitute for .
Step 5.4
Find the integration factor .
Step 6
Step 6.1
Since is constant with respect to , move out of the integral.
Step 6.2
Since is constant with respect to , move out of the integral.
Step 6.3
Multiply by .
Step 6.4
The integral of with respect to is .
Step 6.5
Simplify.
Step 6.6
Simplify each term.
Step 6.6.1
Simplify by moving inside the logarithm.
Step 6.6.2
Exponentiation and log are inverse functions.
Step 6.6.3
Rewrite the expression using the negative exponent rule .
Step 7
Step 7.1
Multiply by .
Step 7.2
Cancel the common factor of .
Step 7.2.1
Factor out of .
Step 7.2.2
Cancel the common factor.
Step 7.2.3
Rewrite the expression.
Step 7.3
Multiply by .
Step 7.4
Apply the distributive property.
Step 7.5
Multiply by .
Step 7.6
Factor out of .
Step 7.6.1
Factor out of .
Step 7.6.2
Factor out of .
Step 7.6.3
Factor out of .
Step 7.7
Factor out of .
Step 7.8
Factor out of .
Step 7.9
Factor out of .
Step 7.10
Rewrite as .
Step 7.11
Move the negative in front of the fraction.
Step 8
Set equal to the integral of .
Step 9
Step 9.1
Apply the constant rule.
Step 9.2
Combine and .
Step 10
Since the integral of will contain an integration constant, we can replace with .
Step 11
Set .
Step 12
Step 12.1
Differentiate with respect to .
Step 12.2
By the Sum Rule, the derivative of with respect to is .
Step 12.3
Evaluate .
Step 12.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 12.3.2
Rewrite as .
Step 12.3.3
Differentiate using the chain rule, which states that is where and .
Step 12.3.3.1
To apply the Chain Rule, set as .
Step 12.3.3.2
Differentiate using the Power Rule which states that is where .
Step 12.3.3.3
Replace all occurrences of with .
Step 12.3.4
Differentiate using the Power Rule which states that is where .
Step 12.3.5
Multiply the exponents in .
Step 12.3.5.1
Apply the power rule and multiply exponents, .
Step 12.3.5.2
Multiply by .
Step 12.3.6
Multiply by .
Step 12.3.7
Raise to the power of .
Step 12.3.8
Use the power rule to combine exponents.
Step 12.3.9
Subtract from .
Step 12.4
Differentiate using the function rule which states that the derivative of is .
Step 12.5
Simplify.
Step 12.5.1
Rewrite the expression using the negative exponent rule .
Step 12.5.2
Combine terms.
Step 12.5.2.1
Combine and .
Step 12.5.2.2
Move the negative in front of the fraction.
Step 12.5.2.3
Combine and .
Step 12.5.2.4
Move to the left of .
Step 12.5.3
Reorder terms.
Step 13
Step 13.1
Solve for .
Step 13.1.1
Move all terms containing variables to the left side of the equation.
Step 13.1.1.1
Add to both sides of the equation.
Step 13.1.1.2
Combine the numerators over the common denominator.
Step 13.1.1.3
Apply the distributive property.
Step 13.1.1.4
Add and .
Step 13.1.1.5
Add and .
Step 13.1.1.6
Cancel the common factor of and .
Step 13.1.1.6.1
Factor out of .
Step 13.1.1.6.2
Cancel the common factors.
Step 13.1.1.6.2.1
Factor out of .
Step 13.1.1.6.2.2
Cancel the common factor.
Step 13.1.1.6.2.3
Rewrite the expression.
Step 13.1.2
Subtract from both sides of the equation.
Step 14
Step 14.1
Integrate both sides of .
Step 14.2
Evaluate .
Step 14.3
Since is constant with respect to , move out of the integral.
Step 14.4
Since is constant with respect to , move out of the integral.
Step 14.5
Multiply by .
Step 14.6
Move out of the denominator by raising it to the power.
Step 14.7
Multiply the exponents in .
Step 14.7.1
Apply the power rule and multiply exponents, .
Step 14.7.2
Multiply by .
Step 14.8
By the Power Rule, the integral of with respect to is .
Step 14.9
Simplify the answer.
Step 14.9.1
Rewrite as .
Step 14.9.2
Simplify.
Step 14.9.2.1
Multiply by .
Step 14.9.2.2
Combine and .
Step 15
Substitute for in .